Current adversarial attacks for multi-class classifiers choose potential adversarial target classes naively based on the classifier's confidence levels. We present a novel adversarial targeting method, \textit{MALT - Mesoscopic Almost Linearity Targeting}, based on local almost linearity assumptions. Our attack wins over the current state of the art AutoAttack on the standard benchmark datasets CIFAR-100 and Imagenet and for different robust models. In particular, our attack uses a \emph{five times faster} attack strategy than AutoAttack's while successfully matching AutoAttack's successes and attacking additional samples that were previously out of reach. We additionally prove formally and demonstrate empirically that our targeting method, although inspired by linear predictors, also applies to non-linear models.